A model of adaptive, market behavior generating positive returns, volatility and system risk
aa r X i v : . [ q -f i n . T R ] S e p A MODEL OF ADAPTIVE, MARKET BEHAVIOR GENERATING POSITIVERETURNS, VOLATILITY AND SYSTEM RISK.
MISHA PEREPELITSAA
BSTRACT . We describe a simple model for speculative trading based on adaptive behavior ofeconomic agents. The adaptive behavior is expressed through a feedback mechanism for changingagents’ stock-to-bond ratios, depending on the past performance of their portfolios. The stock priceis set according to the demand-supply for the asset derived from the agents’ target risk levels.Using the methodology of agent-based modeling we show that agents, acting endogenously andadaptively, create a persistent price bubble. The price dynamics generated by the trading processdoes not reveal any singularities, however the process is accompanied by growing aggregated riskthat indicates increasing likelihood of a crash.
1. I
NTRODUCTION
We consider an agent-based model of economic behavior built on behavioral traits typical inthe models of exchange: a) every agent acts in his/her own interest; b) agents have heterogeneouspreferences; c) agents interact and the interactions result in mutually optimal outcome; d) agents’behavior is adaptive; e) agents share a common belief that it is advantageous to participate in themarket activity. Traits a.–d. describe individual behavior, while the last condition, f., is a groupproperty.The above conditions do not define a model in a unique way, as many different problems canfit into that framework, depending on the meaning of the terms “preferences”, “interactions”, “op-timal”, “own interest”, “adaptive behavior”, and “ an outcome being advantageous”. For theseterms we going to have precise definitions, originating in the analysis of speculative markets, thatwe describe in the paragraphs below. In short, the interactions will mean setting new price of anequity, agents’ preferences will be defined by the stock-to-bond ratio, and adaptive behavior willbe expressed through the changes in stock-to-bond ratios.We will show that when the fundamental value of the traded asset is not readily estimated,being, for example, of speculative nature, the group of agents acting endogenously, according toconditions a.–f. can generate price dynamics with positive return. As there are no exogenousinputs in the model, the outcome is clearly a price bubble resulting from adaptive, self-centeredbehavior of agents. The phenomenon has some interesting features that make it different from thespeculative bubbles due to actions of “chartists” traders, expected utility maximizers, or due to theherding behavior of traders.For one thing, the price bubble is persistent, as all agents in the group find it beneficial to carryon, provided they ignore mounting “systematic risk” of a breakdown (crash) of the market activity.We note here, that the systematic risk is a group property and it is typically not observed by the
Date : September 26, 2018.Email: [email protected]. agents, and thus not factored into the agents’ short-term decision making process. On a long runthe agents will eventually take the risk into consideration, and the price bubble will terminate. Insection 3 we briefly discuss different scenarios of how that process may proceed.Additionally, the model produces price fluctuations with some statistical similarity to fluctu-ations of real markets. A price chart generated by the adaptive model is a graph growing, ordecreasing, at a constant rate with some persistent but not increasing volatility. Charts like that arenot uncommon for the market indexes over a span of few years. We conjecture that the adaptivetype of behavior described in this paper contributes substantially to long term growth dynamics inspeculative markets.Models of adaptive economic behavior were developed by many authors in many different con-texts. In a strategic decision making they were advanced as models of convergence by Cross [3],models of learning in games by Fundenberg-Levine[4], reinforcement learning models by Roth-Erev [17], to mention just a few contributions.The structure of agents’ interactions in the model that we consider in this paper differs from themodels in the above-mentioned references. Here, agents use a singe strategy, or to be more precise all agents use the same rule for the change of their strategies, and non-equilibrium price dynamicsis generated by heterogeneity of agents’ strategies caused by the updating rule. In our study weadopt the methodology of multi-agent dynamic simulations in the spirit of works of Stigler [18],Kim-Markowitz [14], Arthur-Holland-LeBaron-Palmer-Tayler [1], Levy-Levy-Solomon [8].1.1.
Motivation.
The importance of the behavioral aspects of speculative markets has alwaysbeen acknowledged. It is hardy deniable that irrationality of market participants and “animal spir-its” are ubiquitous features that drive price instabilities and critical events. Attempts to understandthe price changes in stock markets lead to the introduction of agent-based models that mimic thebehavioral patterns of different groups of traders. Terms like “fundamentalists”, “noise-traders”,“portfolio re-balancers”, “portfolio insurers”, “adaptive learners” have been introduced into themodels. Kim-Markowitz [14] have shown the destabilizing effect of constant portfolio insurers onthe price dynamics and proposed it as an explanation for the market crush of 1987. The analysis ofmulti-agent models in Levy-Levy-Solomon [5, 6] shows that the rational traders (fundamentalists)do not out-evolve the irrational traders (chartists), and the later group will cause the price dynam-ics to go through a series of booms and crushes, that has statistical similarities with the empiricaldata. A number of non-linear differential equation models were proposed in Lux [9, 10, 11] andLux-Marchesi [12, 13] to describe the evolution of the population of different types of traders,their interactions, the price dynamics and changes in agents’ wealth. The model generates com-plex chaotic dynamics with the clustering of volatility of returns, fat tails in the distribution ofreturns and wealth, as well as the long-term memory. Even if one restricts the traders to be ra-tional, expected utility maximizers, the price will persistently develop so called rational bubblesintroduced in Mandelbrot [15] and Blanchard-Watson[2]. Thus, the significant deviation from thefundamental value is the generic property of speculative prices.This paper focuses on the behavior of the market as a whole, consisting of many kinds stocks.The statistical properties of market portfolios are different from that of a single stock. Lookingat index charts, from after the Great Depression and onward, it is not unusual to see long periodsof the scale 5-10 years when the market indexes grow at nearly constant rates, with mild, non-increasing volatility, and returns that fluctuate randomly around the mean.
DAPTIVE MARKET BEHAVIOR 3
The positive returns, admittedly, are influenced by macro-economic parameters, such as thereturn on US government bonds. But the quantitative expression for the dependency or for the de-viations is not clear. We suggest that some of the features of ”the happy growth” market dynamicscan be generated endogenously by market participants following adaptive strategies. Moreover,we suggest that such behavior contributes to dramatics events that inevitably end the period of op-timistic growth. To provide explanations we analyze an agent-based model in which interactionsbetween agents mimic actual trading processes. The model is built on the following assumptions.In the changing market environment traders constantly have to revise their investment portfolios,moving funds between ”safer” and ”riskier” assets, and to change their risk levels. We will assumethat this re-balancing and updating of portfolios completely determines the dynamics of the marketas a whole. To be more specific we postulate that a) the price of the market portfolio is set byagents willing to re-balance funds between their stock and a safe, cash or bond, asset; b) whenre-balancing, agents act solely on the basis of their stock-to-bond ratios; c) agents change theirinvestment ratios adaptively to the changes in their portfolios.The purpose of this paper is to understand the market dynamics described by the rules a.-c.,isolating this type of behavior from other market factors. The main finding is a formula (4) on page5 for the mean return in the excess of the safe investment return r as a function of the systematicrisk, that is, the risk of a crash, imposed by the collective behavior of the group.2. T HE MODEL
To introduce the model we start with a market of only two agents trading shares of a single asset.The agents, call them Petra and Paula, do not know the asset fundamental value, but expect that theprice can grow at moderate rates for long periods of time, and they always prefer increasing valueto decreasing.The agents meet at time intervals that we label with n = , , .. At period n , Petra’s portfoliohas ( s n , b n ) , dollar value of stock and bond investments, and Paula’s: ( s n , b n ) . The stock price pershare at period n is P n . The next period investment portfolio ( s n + i , b n + i ) and price P n + determinedtrough the following steps.Evaluating the last change in their portfolios, agents set target stock-to-bond ratios for the nextperiod, that we denote by k n + i . The new price P n + is set in such a way that the dollar amountof funds that Petra wishes to move from stocks to bonds, to be at the target ratio k n + , equalsthe dollar amount Paula wants to move from bonds to stock to be at her ratio k n + . This balanceexpressed as(1) P n + P n s n − k n + b n = k n + b n − P n + P n s n . Here we assuming that any fractional amount of a share can be exchanged, and for simplicityof the exposition, consider the case with zero interest in the bond account. With the newly setprice the agents re-balance portfolios and evaluate the market performance by comparing ratios P n + s ni / ( P n b ni ) to target values k n + i . If P n + s ni / ( P n b ni ) > k n + i , from the agent i point of view, themarket performed better than expected, and if P n + s ni / ( P n b ni ) < k n + i it performed worse. Sincewhen one agent is selling shares the other one is buying, Petra and Paula have different opinions MISHA PEREPELITSA on the market performance. The agents update the target portfolio ratios according to the rule(2) k n + i = α k n + i P n + s ni P n b ni > k n + i , k n + i P n + s ni P n b ni = k n + i , β k n + i P n + s ni P n b ni < k n + i . With α > , < β < , we refer to this rule as an adaptive feedback mechanism . If the agentidentifies a growing market she increases her stock-to-bond ratio by a fixed amount, while theother agent decreases her ratio. The feedback reflects the fact that when faced with a series ofbad investments the agent will reduce the equity part of the portfolio, while with the investmentgrowing better than expected, the agent will take riskier position.It easy to see that there is an equilibrium when no shares are traded and price doesn’t change: s ni = s i , b ni = b i , k ni = s i / b i , P n = P . Small deviations will set off a non-trivial dynamics. In thisprocess, Petra and Paula will alternatively change their preferences by factors α and β , so that, onaverage, the stock-to-bond ratios change at the rate ( αβ ) / per period. The gross return, P n + / P n , after a short transient period will settle at the average value of ( αβ ) / per period, but will keeposcillating around that value, see the top graph in Figure 1. The stock price goes up when agentshave bias toward the increase of the risk preferences, expressing the optimistic outlook, quantifiedby the condition αβ > . The story of this process is simple: Petra and Paula are trading stock and and move their stock-to-bond ratios up and down with an upward average trend. This drives up the average demand andconsequently the price.Figure 2 shows a simulated, traded dollar value of stock per period. The volume keeps growingup to a certain limit – the amount of cash available to agents. At the same time, as the stock priceincreases exponentially, the number of traded shares will approach zero.To make a model more viable we can arrange the trading mechanism so that agents do not needto interact directly or even know about each other. They might be matched by a third party, amarket maker, to whom they submit their order books. Moreover, the stock price might changein between the periods when Petra and Paula are trading, due to, for example, actions of otheragents. The middle graph in Figure 1 shows a realization of the stock price when it is changing asa geometric random walk between Petra and Paula trading sessions. The outcome is still the same:Petra and Paula are pushing price up, The rate of increase, per unit of time, depends now on howmany times during that periods they trade.To make the model completely endogenous we populated the market by many copies of Petraand Paula. At every trading period there will be a group of ”active” traders willing to re-balancetheir portfolios. They submit the book orders to the market maker who executes the orders andsets the new price, according to the rule similar to (1), see section 4 for details. Each active agentuses rule (2) to update her stock-to-bond ratio taking into account only the performance of herportfolio. The process proceeds to the next step, with a new, randomly chosen group of active
DAPTIVE MARKET BEHAVIOR 5 traders. The model is determined by the following parameters: N –number of agents, m –number ofactive agents that is fixed or can be chosen from a distribution, τ –the number of trading periods peryear, the feedback mechanism (2) and the set of initial portfolios together with the starting price P per share. With such parameters, each agent, on average, re-balances his portfolio m τ / M timesper year, which we set to be of order 1.A typical realization of price dynamics is shown in Figure 3. Statistically, the returns quicklymove to a stationary regime with no significant long term correlations. The bottom graph in figures1 shows the realizations of the gross return. The product αβ still determines if the stock price ispumped up or down. The formula for the mean return per year(3) E [ R ] / r = ( αβ ) m τ N , where r is the return on cash account, appears to be a good quantitative approximation for thegeometric mean of return on the stock investment, see Perepelitsa-Timofeev [16] for a technicalreport on numerical simulations of the model.3. D ISCUSSION
Besides the price charts, agent-based models provide detailed information on the wealth distri-bution among agents. This information allows one to study the effects of market activity on thepopulation of agents. As an example, let us consider trading between N =
500 agents with theparameters used in simulation presented in Figure 3. The stock price is going up at the rate 14%per year with the standard deviation of 0 . . In 10 years, agents that started with the identicalportfolios and stock-to-bond ratios, have their portfolios distributed according to Figure 4.Note that in total money (stock+bond) all agents are better off participating in the market thanholding everything in cash. Moreover, almost a half of them have increased their bond accounts aswell as stock investments. This however comes at the price of all agents adopting more riskier posi-tions, as measured by their stock-to-bond ratios. After 10 years, the mean ratio in the population is5, compared to its initial value of 1/3. The minimum of the ratio over the group is 2. The group be-comes significantly riskier. We can relate this risk to the systematic risk of the market breakdown.Indeed, every trading period, a real-life agent would face a decision to continue trading using thestrategy (2) or start selling the stock, in hope to recover its cash value. The outcome would dependon what other agents are doing. It can be expressed by a version of a Prisoner’s Dilemma, Table 1,in which the agent plays against the rest of the group and chooses between “staying”, continuingusing adaptive strategy, or “selling,” selling of all stock shares. As the payoffs in the game growthe agent will eventually prefer to use his dominant strategy “sell.” With the rest of the agentsfollowing the same path, the market would crash.The annual growth rate of the mean of the stock-to-bond ratios can be estimated in terms of theparameters of the model as ( αβ ) m τ N . With this we can interpret (3) as(4) E [ R ] / r = the rate of increase of the risk of a market breakdown;the risk premium of the market portfolio is proportional to the rate of increase of the systematicrisk. Interestingly, a similar formula, relating the return to the increasing risk of a crash, applies toa rational bubble, described in Blanchard-Watson [2].As an alternative scenario to a crash, the group may switch to a different feedback mechanism (2)with a ”pessimistic” bias, determined by the condition αβ < . The bubble will deflate gradually
MISHA PEREPELITSA trading periods g r o ss r e t u r n trading periods g r o ss r e t u r n
500 520 540 560 580 600 620 640 660 680 700 trading periods g r o ss r e t u r n F IGURE
1. Gross returns during 200 trading periods. Top figure: deterministic2-agent model. Middle figure: 2-agent model with noise; graph shows only thereturns from Petra and Paula trading sessions. Bottom figure: endogenous, multi-agent model with N =
500 agents and m =
10 active traders. The bottom chartshows 200 trading periods, after returns become stationary. In all models α = . , β = . . Red lines are the geometric mean returns computed from (3): 1.012,1.012 and 1.0006 for the top, middle and bottom chart, respectively.to lower price levels. It can be deflated all the way to the starting price P and the only result ofthis trading cycle will be the redistribution of initial funds among agents.Gradual deflation is certainly a better alternative to a crash, as the former is always accompaniedby a panic behavior that has adverse effects on the economy. Interestingly, from the group prospec-tive, both a crash and a deflation trading might have similar end results. Figure 5 shows simulateddistributions of portfolios after a crash and a 2-year deflation trading. The scatter plots of both DAPTIVE MARKET BEHAVIOR 7 trading periods t r ad i ng v o l u m e , $ F IGURE
2. Dollar value of shares traded per period.
500 520 540 560 580 600 620 640 660 680 700 trading periods l og o f t he s t o ck p r i c e F IGURE
3. Logarithm of the stock price, after the returns become stationary.Straight line is the mean return line of slope (3).distributions are clearly comparable. However, to deflate the bubble with the adaptive feedback (2)requires agents, from time to time, to buy shares when the stock price is clearly decreasing. Thiscondition might go counter with primarily individualistic preferences of agents, leading to a crash.Let us finally mention that when the bubble is considered as a part of a bigger market environ-ment there is also a possibility for it turning into a kind of a self-aggregated Ponzi scheme. Afterall, a long-term stable growth of the bubble will attract new participants. While the influx of newmoney keeps growing at the same rate as the return generated by the bubble, the agents can realizethe profit by selling stocks without dropping the price. The price bubble will prolong its existencebut its fate will be determined by the balance of in- and out-flux of money.
MISHA PEREPELITSA $ investment in bond in 10 year $ i n v e s t m en t i n s t o ck i n y ea r F IGURE
4. Scatter plot of agents’ portfolios in 10 years. Lines of slope -1 are thelines of the constant total wealth. The initial portfolios of all agents is marked with’*’. The mean portfolio is marked with ’x’.MarketAgent stay sellstay b + . s E [ R ] n b + s E [ R | n b T ABLE
1. Schematic decision matrix for Agent vs. Market game with the payoffsto the agent in n years : b is value of the agent’s bond account. s E [ R ] n is thestock account of the agent. The agent recovers all of it in sell-stay scenario. Thepayoff stay-stay is uncertain; the agent expecting to recover cash value for at leasta fraction of her stock investment.4. A PPENDIX : P
ROPERTIES OF THE MODEL
Consider a set of N agents, described by their portfolios ( s i , b i ) , i = . . . N , of dollar values of astock and cash accounts, and let k i stand for agent i stock-to-bond ratio, either actual, or a future,target value. P will denote a current price per share and P the new price determined by agents’demand. Let { i l : l = . . . m } be the set of “active” agents, i.e. the ones setting the new price. Theset of active traders is determined each trading period by a random draw from the population.If x i l is the dollar amount that agent i l wants to invest in stock, then PP s i l + x i l b i l − x i l = k i l . DAPTIVE MARKET BEHAVIOR 9 $ investment in bond $ i n v e s t m en t i n s t o ck F IGURE
5. Scatter plot of agents’ portfolios in 10 years of a growing bubble fol-lowed by a crash or a gradual deflation: ‘o’ represent portfolios after a crash to thestarting level of stock price P ; ‘+’ represent portfolios after 2 years of deflatingbubble dynamics that brings the price down to P ; ‘*’ – portfolios of agents at time t = . The demand-supply balance is m ∑ l = x i l = , which can be solved for P : PP = m ∑ l = k i l b i l + k i l ! m ∑ l = s i l + k i l ! − . During the trading, the total amount of all cash accounts is conserved, as well as the number ofshares of stocks owned by agents. Note here, that we are assuming that any fractional amount ashare can be traded. Once the price is set, agents move corresponding amounts between cash andstock accounts, re-balancing their portfolios. The update mechanism for new stock-to-bond ratiosis given by (2). The interaction is repeated the following trading periods with randomly selectedsets of active agents. As in 2-agent model, there is an equilibrium solution, when all agents havebalanced portfolios, s i / b i = k i , i = , . . . , N , and stock price doesn’t grow, P n = P n − . If money insaving accounts grow at the rate r , the equilibrium price will grow at the same rate P n = rP n − . When the initial data are out of the equilibrium, the system will exhibit non-trivial dynamics,diverging from the equilibrium.In the non-equilibrium regime, the stock price increases or decreases according, together withthe mean stock-to-bond ratios. The return, however, becomes nearly stationary , with the geometricmean approximately equal to (3), see [16]. The distribution of returns depends on the mechanismfor selecting a random group of active agents. If the number of active agents, m , is fixed the positive returns negative returnsm=5 0.1m=10 0.07 -0.09m=50 0.21 -0.47S&P500 0.22 -0.54T ABLE
2. Correlation between mean positive (negative) returns and standard de-viations. The adaptive model with N =
500 agents and m = , ,
50 active traderswas used to compute the mean returns and standard deviations for 1000 differentvalues of α , β ∈ [ . , . ] . For S&P500, 85 pairs of the mean daily returns and stan-dard deviations computed from non-overlapping, semi-annual periods of the indexfrom the period 1960-2010.returns in the stationary regime follow log-normal distribution. If number m is itself random, forexample chosen from an uniform distribution, the returns have significantly more mass at the tailsand at 1, than a corresponding log-normal distribution. Moreover, the returns show no significantcorrelations over the lags of two or more trading periods, while the next period return is negativelycorrelated with the present one, see [16] for details.Of interest are also statistical properties of the price dynamics for a collection of processes withdifferent values of parameters ( α , β ) . Table 2 presents the correlation between mean returns andstandard deviations. For each pair of values we compute the stationary mean positive or negativereturn and the standard deviation, and consider the distribution of these values for a large numberof different values α , β in the vicinity of 1 . The numbers do depend on other parameter of themodel, such as m and N , but they indicate positive correlation between mean positive return and thestandard deviation, and negative correlation between mean negative return and standard deviations.This property is observed in the returns of real markets.R EFERENCES [1] W.B. Arthur, J.H. Holland, B. LeBaron, R.G. Palmer and P. Tayler. AssetPricingunderEndogenousExpecta-tionsinanArtificialStockMarket.in W.B. Arthur, S. Durlauf and D. Lane. eds. The Economy as an EvolvingComplex System II. Addiso-Wesley, 1997.[2] O.J. Blanchard and M.W. Watson. Bubbles, rational expectations and speculative markets. in
Crisis in Eco-nomic and Financial Structure: Bubbles, Bursts and Shocks,
P. Wachtel, editor, Lexington Books, Lexington,MA, 1982.[3] J.G. Cross. Atheoryofadaptiveeconomicbehavior. Cambridge Univ. Press, 1983.[4] D. Fundenberg and D.K. Levine. ThetheoryofLearninginGames. The MIT Press, 1998.[5] M. Levy, H. Levy, and S. Solomon. A microscopicmodel of the stock market: Cycles, booms, and crashes.Economics Letters, 45:103111, 1994.[6] M. Levy, H. Levy, and S. Solomon. Microscopic simulation of the stock market: The effect of microscopicdiversity. Journal de Physique I, 5:1087 1107, 1995.[7] M. Levy, H. Levy, and S. Solomon. New evidence for the power law distribution of wealth. Physica A,242:9094, 1997.[8] M. Levy, H. Levy and S. Solomon. MicroscopicSimulationofFinancialMarkets: FromInvestorBehaviortoMarketPhenomena. Academic Press, 2000.
DAPTIVE MARKET BEHAVIOR 11 [9] T. Lux. Herdbehavior,bubblesandcrashes. Economic Journal, 105:881 896, 1995.[10] T. Lux. The stable Paretian hypothesis and the frequency of large returns: An examination of major Germanstocks. Applied Financial Economics, 6:463475, 1996.[11] T. Lux. The socio-economicdynamics of speculative markets: Interacting agents, chaos, and the fat tails ofreturndistributions. Journal of Economic Behavior and Organization, 33:143165, 1998.[12] T. Lux and M. Marchesi. Scalingandcriticalityinastochasticmulti-agentmodelofafinancialmarket.Nature,397:498500, 1999.[13] T. Lux and M. Marchesi. Volatilityclusteringinfinancialmarkets: Amicro-simulationofinteractingagents.International Journal of Theoretical and Applied Finance, 3:67702, 2000.[14] G. Kim and H. M. Markowitz. Investmentrules, margin and market volatility. Journal of Portfolio Manage-ment, 16:4552, 1989.[15] B. Mandelbrot. Nonlinearforecasts,rationalbubbles,andmartigales. Journal of Business 39, 242-255, 1966.[16] M. Perepelitsa and I. Timofeev. Asynchronousstochasticpricepump, arXiv[17] A.E. Roth and I. Erev. Learninginextensive-formgames: experimentaldataandsimpledynamicsmodelsintheintermediateterm. Games and Economic Behavior, 8 164–212, 1995.[18] G.J. Stigler. PublicRegulationoftheSecuritiesMarket. Journal of Business, 37, 1964M
ISHA P EREPELITSA , MISHA @ MATH . UH . EDU U NIVERSITY OF H OUSTON , PGH 631, 4800 C